Subsurface Causing Mesh Problems
I downloaded the model and found your problem. You've got hundreds or vertices on top of each other.



What you want to do is select everything live I've done and press W to open the specials menu and press "Remove doubles".



As you can see, you've lost all your sharpness in your mesh. You we're probably extruding faces to make these infinitely thin ones that gave you sharp edges, but there's a better way.

What you want to do when you find an edge you want to sharpen it select it and by pressing Ctrl+E to go into edge crease mode and sharpen edges by pulling your cursor from the edge's origin.


Here you can see I hot a little too excited and did a bunch of edges.